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thanks for reading. edited based on helpful comments

Using GeoGebra to add sine waves with the same amplitude together to produce a graph of superposition is able to generate a pseudo-random pattern

I would now like to express this more formally.

I would like to prove that given an infinite set of sine waves, with different frequencies, but with the same amplitude, when summed to create a function, will have a rate of change that always produces minimums.

Reason being I want to prove that the set of pseudo-randomly occurring minimums is also infinite.

The sum of the functions is this;

$ f(x) = \sum_1^\infty [sin^2( \frac x2 \pi) + sin^2( \frac x{2n+1} \pi)] $

and hence the rate of change is $ f'(x) $

Alex Evans
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  • First of all, what kinds of pairs of frequencies are you using? If the frequencies are rational multiples of each other, then the pattern will eventually repeat. If not, we have a more interesting behavior that I would say is closer to "pseudorandom" – Ben Grossmann Aug 01 '22 at 17:27
  • It depends on the frequencies that you use. A more general question: https://math.stackexchange.com/questions/661638/period-of-sum-of-sinusoids – VTand Aug 01 '22 at 17:29
  • Second, to express your observation formally, you need to formally state what observation it is that you're trying to explain. What exactly does "produce constructive and destructive interference all the way along the x-axis to infinity" mean? I suspect that what you ultimately have in mind is something like "the values of $x$ where $f(x)$ attains its maximum value (or gets arbitrarily close) is infinite and unbounded" and similarly for the set of $x$ where $f(x) = 0$. – Ben Grossmann Aug 01 '22 at 17:30
  • yes, good point Ben - the pattern will repeat unless the graphs are augmented by 'n' in some way. For simplicity I just used some simple examples but maybe I over simplified? – Alex Evans Aug 03 '22 at 18:28
  • From your second point I can see that I've definitely over simplified, sorry - the actual plot is $ f''(x) = \sum sin^2( \frac x{2n-1} \pi) + sin^2( \frac x{2} \pi)$ and the minimums of the double differentiation are pseudo random. – Alex Evans Aug 03 '22 at 18:34

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Here's a statement that I think comes close to what you're trying to say about "constructive interference".

Claim: Let $\alpha,\beta \neq 0$ be such that $\alpha/\beta$ is irrational. Let $f:\Bbb R \to \Bbb R$ be given by $f(x) = \sin(\alpha x) + \sin(\beta x)$. Let $M$ denote the supremum $M = \sup_{x \in \Bbb R} f(x)$. For any $\epsilon > 0$, let $S_\epsilon$ denote the set of $x$ for which $f(x) > M- \epsilon$.

$M = 2$, and for any $\epsilon > 0$, the set $S_\epsilon$ is infinite and unbounded.

Sketch of proof: First, we note that $M \leq 2$, since $$ M = \sup_{x \in \Bbb R}(\sin \alpha x + \sin \beta x) \leq \sup_{x \in \Bbb R}\sin\alpha x + \sup_{x \in \Bbb R} \sin \beta x \leq 1 + 1 = 2. $$

We note that $\sin(\alpha x)$ reaches its maximum value at $x_n = \frac{(2n + 1) \pi}{2 \alpha}$ for integer values of $n$. It suffices to show that for any $0 < \epsilon \leq 1$, there exist infinitely many $n \in \Bbb Z$ such that $\sin(\beta x_n) > 1 - \epsilon$. Let $\langle x\rangle$ denote the integer-part of $x$, i.e. $\langle x\rangle = x - \lfloor x \rfloor$. It can be shown that the sequence $\langle \frac{x_n}{2 \pi \beta} \rangle $ is dense in $[0,1]$ (see this post and this post, for instance). From there, it follows that there are infinitely many $n$ for which $$ \frac{\arccos(1 - \epsilon)}{2 \pi} < \frac{x_n}{2 \pi \beta} < \pi - \frac{\arccos(1 - \epsilon)}{2 \pi}, $$ and the conclusion follows.

We could show that $f$ also comes arbtirarily close to its infimum by noting that $f(-x) = -f(x)$ and applying the above result.

Ben Grossmann
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